3 SPECTRUM OF A COMMUTATIVE BANACH ALGEBRA 329 veE(p) and \u(x)\ — 1 almost everywhere. Deduce that we !(//,) and then use (f).) (i) Let /e K(p) be such that (fu dp = 0 for all u e I(/i) for which f u dp, ^ 0. Show that/is /x-negligible. (By applying the canonical decomposition of (d) above to the function 1 + z/, where z is a complex constant, and by using Problem 6 of Section 13.12, show that 1 + zfe E(p,)i then apply Problem 17 of Section 13.21.) (j) Show that every real function /e K(/<0 is equivalent to a constant. (Reduce to the case where \fdp, — 0, and apply (i).) (k) Deduce from (j) that if B denotes the set of functions u such that u e B, then B + B is dense in &p(p,) for 1 ^p < + oo. (If (\lp) + (l/<?) = 1 and if /e ^(p) is such that I fu dp, = 0 and /w cfyx = 0 for all u e B, observe that \(3%f)u dp, > 0, and use (j).) 14. With the notation and hypotheses of Problem 8, assume in addition that there exists only one representative measure p for ^. Let v be a quasi-representative measure for X, and write v = v' + v", where v' is a measure with base p., and v" is disjoint from ju,. Then vf and v" are quasi-representative measures for ^, and c6/' = 0 ("theorem of F. and M. Riesz"). (Assume that \dv ^ 0, by replacing v by v + p* if necessary. If p = |v| = |y'| + IT/'], then D^p) >0 and therefore D^p) = 3n(\h\), where vf=^h-^ by virtue of Problem 1 l(d). Hence there exists a sequence (vn) in B such that \(vn) — I for all n and such that \\vn\ dp~*3n(\h\). Since JM(|/z|) g | |i;n/r| rf/t, the integrals f |yrt/z| dp, also tend to J^(|A|), and | |on| d\if\ -> 0. Write /z =/^, where /e E(ju) and |/| = |/z|, so that g e ^°°(jLt) and \g\ = 1; as in Problem 13(a), show that fvn tends to \fdp, in the mean (with respect to p,). From the hypothesis that v is quasi-represen- tative, deduce that g e I(p), and apply Problem 13(f).) 15. With the hypotheses and notation of Problem 8, suppose that p is a hyperextremal representative measure for %. For 1 "^p < + oo, let ^p(p) denote the set of functions /e &p(p) such that for all #eK(/u,)n J?«(JA), where (l/p) + (l/^) = 1. Then ^Tp(f>t) <= K(ft), and by virtue of Problem 13(f) we have for 1 < /? ^ + oo. The canonical image of ^p(ju,) in Lp(/x) is denoted by Hp(^) and called the Hardy space. (a) For 1 g; p < + oo , show that ^p(jLt) is the closure of B in ^p()u). When ^=+00, ^°°(^) is the closure of B in & °°(ft) with respect to the weak topology. (In each case, show that if g e &q(p) is such that | ug dp, = 0 for all w e B, then g e -S^(/x) n K(JU,) and J g dp, — 0, and consequently 1/^^ = 0 for all /euch that x(w/») ~ 1 fc>r all n an<^